Linear Inequations (in Two Variables)
Class 11th Mathematics RS Aggarwal Solution
- Solve the given inequality x + y ≥ 4 graphically in two – dimensional plane.…
- Solve the given inequality x – y ≤ 3 graphically in two – dimensional plane.…
- Solve the given inequality y – 2 ≤ 3x graphically in two – dimensional plane.…
- Solve the given inequality x ≥ y – 2 graphically in two – dimensional plane.…
- Solve the given inequality 3x + 2y 6 graphically in two – dimensional…
- Solve the given inequality 3x + 5y 15 graphically in two – dimensional…
- Solve the given inequalities x ≥ 2y, y ≥ 3 graphically in two – dimensional…
- Solve the given inequalities 3x + 2y ≤ 12, x ≤ 1, y ≥ 2 graphically in two –…
- Solve the given inequalities x + y ≤ 6, x + y ≥ 4 graphically in two –…
- Solve the given inequalities 2x + y ≥ 6, 3x + 4y ≤ 12 graphically in two –…
- Solve the given inequalities x + y ≤ 9, y x, x ≥ 0 graphically in two –…
- Solve the given inequalities 2x – y 1, x – 2y 1 graphically in two –…
- Solve the given inequalities 5x + 4y ≤ 20, x ≥ 1, y ≥ 2 graphically in two –…
- Solve the given inequalities 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0…
- Solve the given inequalities 2x + y ≥ 4, x + y ≤ 3, 2x – 3y ≤ 6 graphically in…
- Solve the given inequalities x + 2y ≤ 10, x + y ≥ 1, x – y ≤ 0, x ≥ 0, y ≥ 0,…
- Solve the given inequalities 4x + 3y ≤ 60, y ≥ 2x, x ≥ 3, x ≥ 0, y ≥ 0…
- Solve the given inequalities x – 2y ≤ 2, x + y ≥ 3, –2x + y ≤ 4, x ≥ 0, y ≥ 0…
- Solve the given inequalities x + 2y ≤ 100, 2x + y ≤ 120, x + y ≤ 70, x ≥ 0, y ≥…
- Solve the given inequalities x + 2y ≤ 2000, x + y ≤ 1500, y ≤ 600, x ≥ 0, y ≥ 0…
- Solve the given inequalities 3x + 2y ≥ 24, 3x + y ≤ 15, x ≥ 4, graphically in…
- Solve the given inequalities 2x – y ≤ –2, x – 2y ≥ 0, x ≥ 0, y ≥ 0 graphically…
- Solve the given inequalities 3x + y ≥ 12, x + y ≥ 9, x ≥ 0, y ≥ 0. graphically…
- Find the linear inequalities for which the shaded area is the solution set in…
- Find the linear inequalities for which the shaded area is the solution set in…
- A furniture dealer deals in only two items : tables and chairs. He has 30000 to…
- If a young man rides his motorcycle at 40 km per hour, he has to spend 6 per km…
Exercise 7
Question 1.Solve the given inequality x + y ≥ 4 graphically in two – dimensional plane.
Answer:
The graphical representation of x + y ≥ 4 is given by blue line in the figure below.
This lines divides x-y plane into two parts
Select a point (not on the line),which lies on one of the two parts, to determine whether the point satisfies the given inequality or not.
We select the point as (0,0)
It is observed that 
Therefore, the solution for the given inequality including the points on the line.
This can be represented as follows,
Question 2.
Solve the given inequality x – y ≤ 3 graphically in two – dimensional plane.
Answer:
The graphical representation of x – y ≤ 3 is given by blue line in the figure below.
This lines divides x-y plane into two parts .
Select a point (not on the line),which lies on one of the two parts, to determine whether the point satisfies the given inequality or not.
We select the point as (0,0)
It is observed that 
Therefore, the solution for the given inequality including the points on the line.
This can be represented as follows,
Question 3.
Solve the given inequality y – 2 ≤ 3x graphically in two – dimensional plane.
Answer:
The graphical representation of y – 2 ≤ 3x is given by blue line in the figure below.
This lines divides x-y plane into two parts .
Select a point (not on the line),which lies on one of the two parts,to determine whether the point satisfies the given inequality or not.
We select the point as (0,0)
It is observed that 
Therefore, the solution for the given inequality including the points on the line.
This can be represented as follows,
Question 4.
Solve the given inequality x ≥ y – 2 graphically in two – dimensional plane.
Answer:
The graphical representation of x ≥ y – 2 is given by blue dotted line in the figure below.
This lines divides x-y plane into two parts .
Select a point (not on the line),which lies on one of the two parts,to determine whether the point satisfies the given inequality or not.
We select the point as (0,0)
It is observed that 
Therefore, the solution for the given inequality excluding the points on the line.
This can be represented as follows,
Question 5.
Solve the given inequality 3x + 2y > 6 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by blue dotted line in the figure below.
This lines divides x-y plane into two parts .
Select a point (not on the line),which lies on one of the two parts,to determine whether the point satisfies the given inequality or not.
We select the point as (0,0)
It is observed that 
Therefore, the solution for the given inequality excluding the points on the line.
This can be represented as follows,
Question 6.
Solve the given inequality 3x + 5y < 15 graphically in two – dimensional plane.
Answer:
The graphical representation of 3x + 5y < 15 is given by blue dotted line in the figure below.
This lines divides x-y plane into two parts .
Select a point (not on the line),which lies on one of the two parts,to determine whether the point satisfies the given inequality or not.
We select the point as (0,0)
It is observed that 
Therefore, the solution for the given inequality excluding the points on the line.
This can be represented as follows,
Question 7.
Solve the given inequalities x ≥ 2y, y ≥ 3 graphically in two – dimensional plane.
Answer:
The graphical representation of x ≥ 2y, y ≥ 3 is given by common region in the figure below.
.…… (1)
……. (2)
Inequality (1) represents the region below line x=2y(including the line x=2y).
Inequality (2) represents the region above line y=3(including the line y=3).
Therefore, every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 8.
Solve the given inequalities 3x + 2y ≤ 12, x ≤ 1, y ≥ 2 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region behind line 
(including the line x=1).
Inequality (3) represents the region above line 
(including the line y=2).
Therefore, every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 9.
Solve the given inequalities x + y ≤ 6, x + y ≥ 4 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region above line 
(including the line x+y=4).
Therefore, every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 10.
Solve the given inequalities 2x + y ≥ 6, 3x + 4y ≤ 12 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
Inequality (1) represents the region above line 
(including the line ).
Inequality (2) represents the region below line 
(including the line 3x+4y=12).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 11.
Solve the given inequalities x + y ≤ 9, y < x, x ≥ 0 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region below line 
(excluding the line ).
Inequality (3) represents the region in front of line 
(including the line ).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 12.
Solve the given inequalities 2x – y > 1, x – 2y < 1 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
Inequality (1) represents the region below line 
(excluding the line ).
Inequality (2) represents the region above line 
(excluding the line 
).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 13.
Solve the given inequalities 5x + 4y ≤ 20, x ≥ 1, y ≥ 2 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region in front of line (including the line ).
Inequality (3) represents the region above line (including the line ).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 14.
Solve the given inequalities 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region below line 
(including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line 
(including the line ).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 15.
Solve the given inequalities 2x + y ≥ 4, x + y ≤ 3, 2x – 3y ≤ 6 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
Inequality (1) represents the region above line 
(including the line ).
Inequality (2) represents the region below line 
(including the line ).
Inequality (3) represents the region above line 
(including the line ).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 16.
Solve the given inequalities x + 2y ≤ 10, x + y ≥ 1, x – y ≤ 0, x ≥ 0, y ≥ 0, graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
……. (5)
Inequality (1) represents the region below line 
(including the line 
).
Inequality (2) represents the region above line 
(including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line (including the line ).
Inequality (5) represents the region above line 
(including the line ).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 17.
Solve the given inequalities 4x + 3y ≤ 60, y ≥ 2x, x ≥ 3, x ≥ 0, y ≥ 0 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
……. (5)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region above line 
(including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line (including the line ).
Inequality (5) represents the region in front of line 
(including the line )
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 18.
Solve the given inequalities x – 2y ≤ 2, x + y ≥ 3, –2x + y ≤ 4, x ≥ 0, y ≥ 0 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
……. (5)
Inequality (1) represents the region above line 
(including the line 
).
Inequality (2) represents the region above line (including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line (including the line ).
Inequality (5) represents the region below line 
(including the line ).
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 19.
Solve the given inequalities x + 2y ≤ 100, 2x + y ≤ 120, x + y ≤ 70, x ≥ 0, y ≥ 0 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
……. (5)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region below line 
(including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line (including the line ).
Inequality (5) represents the region below line 
(including the line )
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 20.
Solve the given inequalities x + 2y ≤ 2000, x + y ≤ 1500, y ≤ 600, x ≥ 0, y ≥ 0 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
……. (5)
Inequality (1) represents the region below line 
(including the line ).
Inequality (2) represents the region below line 
(including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line (including the line ).
Inequality (5) represents the region below line 
(including the line )
Therefore,every point in the common shaded region including the points on the respective lines represents the solution for the given inequalities.
This can be represented as follows,
Question 21.
Solve the given inequalities 3x + 2y ≥ 24, 3x + y ≤ 15, x ≥ 4, graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
Inequality (1) represents the region above line 
(including the line ).
Inequality (2) represents the region below line 
(including the line ).
Inequality (3) represents the region in front of line 
(including the line ).
Therefore, we can see in the figure that there is no common shaded region.
So there linear inequalities in equations has no solution.
This can be represented as follows,
Question 22.
Solve the given inequalities 2x – y ≤ –2, x – 2y ≥ 0, x ≥ 0, y ≥ 0 graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
Inequality (1) represents the region above line 
(including the line ).
Inequality (2) represents the region below line 
(including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line (including the line ).
Therefore, we can see in the figure that there is no common shaded region.
So there linear inequalities in equations has no solution.
This can be represented as follows,
Question 23.
Solve the given inequalities 3x + y ≥ 12, x + y ≥ 9, x ≥ 0, y ≥ 0. graphically in two – dimensional plane.
Answer:
The graphical representation of 
is given by common region in the figure below.
.…… (1)
……. (2)
……. (3)
……. (4)
Inequality (1) represents the region above line 
(including the line ).
Inequality (2) represents the region above line (including the line ).
Inequality (3) represents the region in front of line (including the line ).
Inequality (4) represents the region above line (including the line )
It is clear from the graph , that the region is unbounded.
Therefore , the following system of inequation is an unbounded set.
This can be represented as follows,
Question 24.
Find the linear inequalities for which the shaded area is the solution set in the figure given below.
Answer:
We have seen that the shaded region and origin are on the same side of the line
For (0,0) we have 
. So the shaded region satisfies the inequality 
.
We have seen that the shaded region and origin are on the same side of the line 
For (0,0) we have . So the shaded region satisfies the inequality 
.
Also , the region lies in the first quadrent . Therefore 
and 
Thus the linear inequation comprising the given solution set are 
, ,
,
Question 25.
Find the linear inequalities for which the shaded area is the solution set in the figure given below.
Answer:
We have seen that the shaded region and origin are on the opposite side of the line 
For (0,0) we have 
. So the shaded region satisfies the inequality 
.
We have seen that the shaded region and origin are on the opposite side of the line 
For (0,0) we have 
. So the shaded region satisfies the inequality 
.
We have seen that the shaded region and origin are on the same side of the line
For (0,0) we have . So the shaded region satisfies the inequality 
.
We have seen that the shaded region and origin are on the same side of the line 
For (0,0) we have 
. So the shaded region satisfies the inequality 
.
Thus the linear inequation comprising the given solution set are 
, ,
,
Question 26.
A furniture dealer deals in only two items : tables and chairs. He has 30000 to invest and a space to store at most 60 pieces. A table costs him 1500 and a chair 300. Formulate the data in the form of inequations and draw a graph representing the solution of these inequation.
Answer:
Let the number of tables and chairs be x and y respectively.
Therefore
Now the maximum number of pieces he can store = 60.
Therefore , 
…….(1)
Also it is given that maximum amount he can invest = 30000
Therefore , 
…… (2)
Therefore , the shaded protion (i.e. A) together with its boundary represents the solution set of the given inequation.
No. of tables = x = 10
No. of chair = y = 50
Question 27.
If a young man rides his motorcycle at 40 km per hour, he has to spend 6 per km on petrol and if he rides it at 50 km hour, the petrol cost rises to 10 per km. He has 500 to spend on petrol and wishes to find the maximum distance he can travel within one hour. Formulate the data in the form of inequation and draw a graph representing the solution of these inequations.
Answer:
Let the distance covered with speed 40 km/hr = x km
And the distance covered with speed 50 km/hr = y km
We know that,
Therefore , maximum speed covered within one hour is
Thus , according to equation ,
Maximum speed covered , 
Subject to the constraint ,
Now plotting both the line on graph paper , we have ,
Distance covered with speed 40 km/hr = x = 0
Distance covered with speed 50 km/hr = y = 50
Therefore , maximum distance covered = 0 + 50 = 50 km